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The fundamental defintion behind areas comes from a square of side 1 unit by 1 unit. The area of such a square is 1 sq units. By this if we say area of any other figure as say 'A' sq units. It means the figure contains 'A' number of squares of area unit square each.
Similarly a rectangle of side lengths 'a' and 'b' can be broken into smaller squares of side length 1 each. There will be a total of 'a' * 'b' such squares.
A right angle triangle can be extended to complete a rectangle with double the area of the triangle. See figure on right side and left side.
Area of right angle triangle = 1/2 * b * h
In case of a general triangle, we can divide it into right angled triangles
Area of general triangle = 1/2 * b * h
In case of a circle, we have an interesting analogy, we can divide circle into small thin strips to form a triangle, area = 1/2 * 2*pi*r = pi*r*r
Alternatively by Integral Calculus, we have area = pi*r*r
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